Is the prayer of the congregant valid if he does not perform the prostration of forgetfulness with the Imam, knowing that the Imam prostrated for a lapse that the congregant did not commit?

You were required to follow the Imam in the prostration of forgetfulness (sujud al-sahw), and since you did not do so, the Shafi'is consider the Imam's sujud al-sahw an essential part (rukn) for the congregant. If the separation was minor, you were obliged to return and perform it; otherwise, the prayer must be repeated. Therefore, you must repeat the Dhuhr prayer mentioned.